Knot Contact Homology is a powerful invariant assigning to each smooth knot in three-space a differential graded algebra. The homology of this algebra is in general difficult to calculate. We will discuss the cord algebra of a knot, which allows us to calculate the grading 0 knot contact homology. We will also see a method of extracting information from augmentations of the algebra.

The mapping class group of a surface is a quotient of the group of orientation preserving diffeomorphisms. However the mapping class group generally can't be lifted to the group of diffeomorphisms, and even many subgroups can't be lifted. Given a surface S of genus at least 2 and a marked point z, the fundamental group of S naturally injects to a subgroup of MCG(S,z). I will present a result of Bestvina-Church-Souto that this subgroup can't be lifted to the diffeomorphisms fixing z.

We'll prove the simplest case of Hirzebruch's signature
theorem, which relates the first Pontryagin number of a smooth 4-manifold
to the signature of its intersection form. If time permits, we'll discuss
the more general case of 4k-manifolds. The result is relevant to Prof.
Margalit's ongoing course on characteristic classes of surface bundles.

Given a vector bundle over a smooth manifold, one can give an alternate
definition of characteristic classes in terms of geometric data, namely connection
and curvature. We will see how to define Chern classes and Euler class for the a
vector bundle using this theory developed in mid 20th century.

Let MCG(g) be the mapping class group of a surface of genus g. For
sufficiently large g, the nth homology (and cohomology) group of MCG(g) is
independent of g. Hence we say that the family of mapping class groups
satisfies homological stability. Symmetric groups and braid groups also
satisfy homological stability, as does the family of moduli spaces of
certain higher dimensional manifolds. The proofs of homological stability
for most families of groups and spaces follow the same basic structure, and
we will sketch the structure of the proof in the case of the mapping class
group.

Consider the Beltrami equation f_{\bar z}=\mu *f_{z}. The prime aim is to
investigate f in its dependence on \mu. If \mu depends analytically,
differentiably, or continuously on real parameters, the same is true for f;
in the case of the plane, the results holds also for complex parameters.

To any compact Hausdorff space we can assign the ring of (classes of)
vector bundles under the operations of direct sum and tensor product. This assignment
allows the construction of an extraordinary cohomology theory for which the long
exact sequence of a pair is 6-periodic.

The aim of this talk is to give fairly self contained proof of the following result due to Eliashberg. There is exactly one holomorphically fillable contact structure on $T^3$. If time permits we will try to indicate different notions of fillability of contact manifolds in dimension 3.